A Generalized Backward Equation For One Dimensional Processes

Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f, the backward Kolmogorov equation gives a condition for f(t,X) to be a local martingale. We generalize the backward equation in two main ways. First, it is extended to non-differentiable functions. Second, the process X is not required to satisfy an SDE. Instead, it is only required to be a quasimartingale satisfying an integrability condition, and the martingale condition for f(t,X) is then expressed in terms of the marginal distributions, drift measure and jumps of X. The proof involves the stochastic calculus of Dirichlet processes and a time-reversal argument. These results are then applied to show that a continuous and strong Markov martingale is uniquely determined by its marginal distributions.